Linear series on a special rational surface

We study a result of Postnikov-Shapiro concerning the Hilbert series of a family of ideals Jφ generated by powers of linear forms in k[x1, . . . , xn]. Using the results of Emsalem-Iarrobino, we formulate this as a problem about fatpoints in P. In the three variable case this is equivalent to studying the dimension of a linear series on a blow up of P. The ideals that arise have the points in very special position, but because there are only seven points, we can apply results of Harbourne to obtain the classes of the negative curves. Reducing to an effective, n.e.f. divisor and using Riemann-Roch yields the desired Hilbert series. Postnikov and Shapiro observe that for a family of ideals closely related to Jφ a result similar to theirs seems to hold. Our methods allow us to prove this for n = 3.